Related papers: Axiomatizing Mathematical Theories: Multiplication
The infinitary propositional logic of here-and-there is important for the theory of answer set programming in view of its relation to strongly equivalent transformations of logic programs. We know a formal system axiomatizing this logic…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial…
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
In this paper, we determine the complexity of the satisfiability problem for various logics obtained by adding numerical quantifiers, and other constructions, to the traditional syllogistic. In addition, we demonstrate the incompleteness of…
We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of…
Independence logic cannot be effectively axiomatized. However, first-order consequences of independence logic sentences can be axiomatized. In this article we give an explicit axiomatization and prove that it is complete in this sense. The…
Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2],…
This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference…
This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into…
Axiomatic approach has demonstrated its power in mathematics. The main goal of this preprint is to show that axiomatic methods are also very efficient for computer science. It is possible to apply these methods to many problems in computer…
Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have…
We investigate the position that foundational theories should be modelled on ordinary computability. In this context, we investigate the metamathematics of $\Sigma$ formulas. We consider theories whose axioms are implications between…
In this paper we will see deductive systems for classical propositional and predicate logic in the calculus of structures. Like sequent systems, they have a cut rule which is admissible. In addition, they enjoy a top-down symmetry and some…
Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…
Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those…
The stipulation that no measurable quantity could have an infinite value is indispensable in physics. At the same time, in mathematics, the possibility of considering an infinite procedure as a whole is usually taken for granted. However,…
Well-founded fixed points have been used in several areas of knowledge representation and reasoning and to give semantics to logic programs involving negation. They are an important ingredient of approximation fixed point theory. We study…
Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…
Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those…